# Институт вычислительной математики

и математической геофизики

# The International Conference on Computational Mathematics

ICCM-2004

June, 21-25, 2004
Novosibirsk, Academgorodok

## Abstracts

*Numerical solution of differential and integral equations*

## Third and Fourth Order's Evolutionary Equations Spliting

**South Ural State University (Chelyabinsk)**

New splitting methods (decomposition, fractional steps, and alternating
directions) are proposed for partial differential evolutionary equatons.
The development of these methods is based on diagonal implicit methods.
For these methods to have the highest attainable order of accuracy,
their coefficients must satisfy both previously known and certain
additional conditions. The coefficients found by using a certain
technique that optimally takes into account stability and accuracy
requirements.

Assume that, in evolutionary equation $u_t=Au+f(t)$
the differential operator $A$ is the sum of operators
$A_{i},quad i=1,dots,p$. Consider also a different decomposition
$A=sum_{i=1}^s A_{i}^o$ $(sge p).$
For example, if $A=A_1+A_2$, and $s=4$ then
$A_1^o=A_3^o=A_1/2$ and $A_2^o=A_4^o=A_2$. We suggest
a new $s$-stage implicit diagonally Runge-Kutta method

$$
u_{n+1}=u_{n}+ au sum_{i=1}^s b_{i} left(Ay_i+f_i
ight),
$$

$$
y_{i}=u_{n}+ au sum_{j=1}^{i-1} a_{ij}left(Ay_j+f_j
ight)+ au a_{ii}
left(A_i^o y_i+g_i+f_i
ight),
$$

where

$$
f_i=fleft(t_n+c_i au
ight),quad
g_1=sum_{j=2}^s A_j^o u,quad
g_i=sum_{j=1,j
e i}^s A_j^o y_j(i>1).
$$

It is possible to find coefficients of methods for $p=2, s=2,3,4$,
$p=3, s=2,3$. The test computations provide better results as compared
to the relaxation and predictor-correct methods. The relaxation and
predictor-correct methods are advantageous in that their implementation
requires fewer arithmetic operations; however, they provide good
resultsonly when the time step is small.

References

1.Shirobokov, N.V., Diagonal Implicit Runge--Kutta
Scemes. Zh. Vychisl. Mat. Mat. Fiz.,
2002, vol. 42, no. 7, pp. 1013--1018.

2.Shirobokov, N.V., Evolutionary Equations Spliting on the Basis
Implicit Methods. Zh. Vychisl. Mat. Mat. Fiz.,
2003, vol. 43, no. 9, pp. 1416--1423.

*Note. Abstracts are published in author's edition*

© 1996-2000, Siberian Branch of Russian Academy of Sciences, Novosibirsk

Last update: 06-Jul-2012 (11:52:06)